Let the distribution of X depend on the parameters (, ) = (1,...,r, 1,...,s). A test of

Let the distribution of X depend on the parameters (θ, ϑ) =

(θ1,...,θr, ϑ1,...,ϑs). A test of H : θ = θ0 is locally strictly unbiased if for each ϕ,

(a) βϕ(θ0, ϕ) = α,

(b) there exists a θ-neighborhood of θ0 in which

βϕ(θ, ϑ) > α for θ = θ0.

(i) Suppose that the first and second derivatives

βi

ϕ(ϑ) = ∂

∂θi

βϕ(θ, ϑ)

%

%

%

%

θ0 and βij

ϕ (ϑ) = ∂2

∂θi∂θj

βϕ(θ, ϑ)

%

%

%

%

θ0 exist for all critical functions ϕ and all ϑ. Then a necessary and sufficient condition for ϕ to be locally strictly unbiased is that β

ϕ = 0 for all i and

ϑ, and that the matrix (βij

ϕ (ϑ)) is positive definite for all ϑ.

(ii) A test of H is said to be of type E (type D is s = 0 so that there are no nuisance parameters) if it is locally strictly unbiased and among all tests with this property maximizes the determinant |(βij ϕ )|.
7 (This determinant under the stated conditions turns out to be equal to the Gaussian curvature of the power surface at θ0.) Then the test ϕ0 given by (7.7) for testing the general linear univariate hypothesis (7.3) is of type E.
[(ii): With θ = (η1,...,ηr) and ϑ = (ηr+1,...,ns, σ), the test ϕ0, by Problem 7.5, has the property of maximizing the surface integral 
S [βϕ(η, σ2 ) − α] dA 
among all similar (and hence all locally unbiased) tests where S = {(η1,...,ηr) : r i=1 η2 i = ρ2σ2}. Letting ρ tend to zero and utilizing the conditions βi ϕ(ϑ)=0, 
S ηiηj dA = 0 for i = j, 
S η2 i dA = k(ρσ), one finds that ϕ0 maximizes r i=1 βii ϕ (η, σ2) among all locally unbiased tests.
Since for any positive definite matrix, |(βij ϕ )| ≤
βii ϕ , it follows that for any locally strictly unbiased test ϕ, |(βij ϕ )| ≤ βii ϕ ≤
Σβii ϕ
r r ≤
Σβii ϕ0 r r = [β11 ϕ0 ]
r = |(βij ϕ0 )|.]